# Differential Equation First Order Linear

• ruiwp13
In summary, the conversation discusses how to solve the differential equation y*e^(x^2)*dy/dx=x+xy and whether the attempted solution is correct. The suggestion is made to factor x+xy into x(1+y) and separate the equation, and it is confirmed that this is the correct approach.
ruiwp13

## Homework Statement

Solve the following differential equation: y*e^(x^2)*dy/dx=x+xy

y'+P(x)*y=Q(x)

## The Attempt at a Solution

I tried to modify the equation to match the first order linear one, and I got:

e^(x^2)*dy/dx=x/y+x (divided everything by y),

but now I get:

dy/dx-x*1/y=e^(-x^2)*x

so instead of the normal form, I have now y'-P(x)*1/y = Q(x).

Did I do something wrong or missing something?

ruiwp13 said:

## Homework Statement

Solve the following differential equation: y*e^(x^2)*dy/dx=x+xy

y'+P(x)*y=Q(x)

## The Attempt at a Solution

I tried to modify the equation to match the first order linear one, and I got:

e^(x^2)*dy/dx=x/y+x (divided everything by y),

but now I get:

dy/dx-x*1/y=e^(-x^2)*x

so instead of the normal form, I have now y'-P(x)*1/y = Q(x).

Did I do something wrong or missing something?

Why don't you just factor x+xy into x(1+y) and separate?

Dick said:
Why don't you just factor x+xy into x(1+y) and separate?

∫y/(1+y) dy=∫e^(-x^2)*x dx

Like this?

ruiwp13 said:
∫y/(1+y) dy=∫e^(-x^2)*x dx

Like this?

Yes, like that.

## 1. What is a first order linear differential equation?

A first order linear differential equation is a mathematical equation that describes the relationship between a dependent variable and its derivative. It is called "linear" because the dependent variable and its derivative appear only in the first degree.

## 2. How do you solve a first order linear differential equation?

To solve a first order linear differential equation, you can use the method of separation of variables or the method of integrating factors. The method of separation of variables involves isolating the dependent variable and its derivative on opposite sides of the equation and then integrating both sides. The method of integrating factors involves multiplying the entire equation by an "integrating factor" to make it easier to solve.

## 3. What is the significance of the "initial condition" in a first order linear differential equation?

The initial condition in a first order linear differential equation is a known value of the dependent variable at a specific point in time or space. It is necessary to include this condition in the equation because it helps to uniquely determine the solution.

## 4. What are some real-life applications of first order linear differential equations?

First order linear differential equations have various applications in physics, engineering, and economics. They can be used to model population growth, radioactive decay, chemical reactions, and electrical circuits. They can also be used to analyze the flow of fluids and the motion of objects under the influence of external forces.

## 5. How do first order linear differential equations differ from higher order differential equations?

First order linear differential equations only involve the dependent variable and its first derivative, while higher order differential equations involve higher derivatives. This means that the solution to a first order linear differential equation will only have one arbitrary constant, while the solution to a higher order differential equation may have multiple arbitrary constants.

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